Dichotomy Theorems for Families of Non-Cofinal Essential Complexity

We prove that for every Borel equivalence relation E, either E is Borel reducible to E₀, or the family of Borel equivalence relations incompatible with E has cofinal essential complexity. It follows that if F is a Borel equivalence relation and F is a family of Borel equivalence relations of non-cofinal essential complexity which together satisfy the dichotomy that for every Borel equivalence relation E, either E∈F or F is Borel reducible to E, then F consists solely of smooth equivalence relations, thus the dichotomy is equivalent to a known theorem.